scholarly journals Uniformly weak differentiability of the norm and a condition of Vlasov

1976 ◽  
Vol 21 (4) ◽  
pp. 393-409 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Smoothness Condition ◽  
Dual Norm ◽  
Weak Differentiability

AbstractIn determining geometrical conditions on a Banach space under which a Chebychev set is convex, Vlasov (1967) introduced a smoothness condition of some interest in itself. Equivalent forms of this condition are derived and it is related to uniformly weak differentiability of the norm and rotundity of the dual norm.

scholarly journals On a Characterisation of Differentiability of the Norm of a Normed Linear Space

1971 ◽  
Vol 12 (1) ◽  
pp. 106-114 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Dual Norm ◽  
Uniform Rotundity ◽  
Strong Differentiability ◽  
Differentiability Properties

The purpose of this paper is to show that the various differentiability conditions for the norm of a normed linear space can be characterised by continuity conditions for a certain mapping from the space into its dual. Differentiability properties of the norm are often more easily handled using this characterisation and to demonstrate this we give somewhat more direct proofs of the reflexivity of a Banach space whose dual norm is strongly differentiable, and the duality of uniform rotundity and uniform strong differentiability of the norm for a Banach space.

scholarly journals BIDUAL OCTAHEDRAL RENORMINGS AND STRONG REGULARITY IN BANACH SPACES

2019 ◽  
pp. 1-17 ◽  
Author(s):  
Johann Langemets ◽  
Ginés López-Pérez
Keyword(s):  
Banach Space ◽  
Studia Math ◽  
Direct Consequence ◽  
Baire Class ◽  
Dual Norm ◽  
Separable Predual ◽  
Separable Case ◽  
Strongly Regular ◽  
First Baire Class

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

On exponential long-time evolution in statistical mechanics

1995 ◽  
Vol 448 (1933) ◽  
pp. 293-319 ◽  
Keyword(s):  
Dynamical System ◽  
Banach Space ◽  
Statistical Mechanics ◽  
Time Correlation ◽  
Smoothness Condition ◽  
Time Correlation Functions ◽  
Time Asymmetry ◽  
Discrete Time Dynamical System ◽  
Long Time ◽  
Non Equilibrium

Penrose & Coveney (1994) recently introduced an invertible discrete-time dynamical system called the pastry-cook’s transformation, for which they constructed a ‘canonical’ non-equilibrium ensemble. In the present paper, we apply the Brussels formalism of non-equilibrium statistical mechanics to this system. The use of the formalism is justified rigorously, and the operators which arise in the theory are calculated exactly. The set of ensembles for which the theory is valid is a Banach space of functions satisfying a certain smoothness condition. This condition ensures that ensembles show a decay towards equilibrium, in agreement with the time asymmetry observed in thermodynamics. We also calculate the decay of time correlation functions using Ruelle’s theory of dynamical resonances. We find that all three methods furnish essentially the same description of the exponential decay to equilibrium in this system.

DUAL DIFFERENTIATION SPACES

2019 ◽  
Vol 99 (03) ◽  
pp. 467-472
Author(s):  
WARREN B. MOORS ◽  
NEŞET ÖZKAN TAN
Keyword(s):  
Banach Space ◽  
Open Problem ◽  
Dense Subset ◽  
Studia Math ◽  
Dual Norm ◽  
Dual Differentiation ◽  
Rotund Norm ◽  
Norm Attaining

We show that if $(X,\Vert \cdot \Vert )$ is a Banach space that admits an equivalent locally uniformly rotund norm and the set of all norm-attaining functionals is residual then the dual norm $\Vert \cdot \Vert ^{\ast }$ on $X^{\ast }$ is Fréchet at the points of a dense subset of $X^{\ast }$ . This answers the main open problem in a paper by Guirao, Montesinos and Zizler [‘Remarks on the set of norm-attaining functionals and differentiability’, Studia Math. 241 (2018), 71–86].

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